I'm trying to find a mathematical way to solve the game Flip It(http://www.coolmath-games.com/0-flipit/index.html). I've reduced to it a system of modular equivalences, but there's too many to solve with substitution. How does one solve a system of modular equivalences?

Attached is the definition of the actually system of modular equivalences in a PDF that I typeset in LaTeX.

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